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A mathematical exercise is a routine application of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
or other mathematics to a stated challenge.
Mathematics teacher In contemporary education, mathematics education, known in Europe as the didactics or pedagogy of mathematics – is the practice of teaching, learning and carrying out scholarly research into the transfer of mathematical knowledge. Although rese ...
s assign mathematical exercises to develop the skills of their students. Early exercises deal with
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
, and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s. Extensive courses of exercises in
school A school is an educational institution designed to provide learning spaces and learning environments for the teaching of students under the direction of teachers. Most countries have systems of formal education, which is sometimes compuls ...
extend such
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
to
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ratio ...
s. Various approaches to
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
have based exercises on relations of
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s, segments, and
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...
s. The topic of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
gains many of its exercises from the
trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
. In college mathematics exercises often depend on
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s of a
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
variable or application of
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s. The standard exercises of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
involve finding
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s and
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
s of specified functions. Usually instructors prepare students with worked examples: the exercise is stated, then a model answer is provided. Often several worked examples are demonstrated before students are prepared to attempt exercises on their own. Some texts, such as those in
Schaum's Outlines Schaum's Outlines () is a series of supplementary texts for American high school, AP, and college-level courses, currently published by McGraw-Hill Education Professional, a subsidiary of McGraw-Hill Education. The outlines cover a wide variety of ...
, focus on worked examples rather than theoretical treatment of a mathematical topic.


Overview

In primary school students start with single digit arithmetic exercises. Later most exercises involve at least two digits. A common exercise in
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entail ...
calls for
factorization In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kin ...
of
polynomial In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s. Another exercise is
completing the square : In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form :ax^2 + bx + c to the form :a(x-h)^2 + k for some values of ''h'' and ''k''. In other words, completing the square places a perfe ...
in a
quadratic polynomial In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomia ...
. An artificially produced word problem is a genre of exercise intended to keep mathematics relevant.
Stephen Leacock Stephen P. H. Butler Leacock (30 December 1869 – 28 March 1944) was a Canadian teacher, political scientist, writer, and humorist. Between the years 1915 and 1925, he was the best-known English-speaking humorist in the world. He is known ...
described this type: :The student of arithmetic who has mastered the first four rules of his art and successfully striven with sums and
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
s finds himself confronted by an unbroken expanse of questions known as problems. These are short stories of adventure and industry with the end omitted and, though betraying a strong family resemblance, are not without a certain element of romance. A distinction between an exercise and a
mathematical problem A mathematical problem is a problem that can be represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more ...
was made by Alan H. Schoenfeld: :Students must master the relevant subject matter, and exercises are appropriate for that. But if rote exercises are the only kinds of problems that students see in their classes, we are doing the students a grave disservice. He advocated setting challenges: :By "real problems" ... I mean mathematical tasks that pose an honest challenge to the student and that the student needs to work at in order to obtain a solution. A similar sentiment was expressed by Marvin Bittinger when he prepared the second edition of his textbook: :In response to comments from users, the authors have added exercises that require something of the student other than an understanding of the immediate objectives of the lesson at hand, yet are not necessarily highly challenging. The
zone of proximal development The zone of proximal development (ZPD) is a concept in educational psychology. It represents the distance between what a learner is capable of doing unsupported, and what they can only do supported. It is the range where they are capable only with ...
for each student, or cohort of students, sets exercises at a level of difficulty that challenges but does not frustrate them. Some comments in the preface of a calculus textbook show the central place of exercises in the book: :The exercises comprise about one-quarter of the text – the most important part of the text in our opinion. ... Supplementary exercises at the end of each chapter expand the other exercise sets and provide cumulative exercises that require skills from earlier chapters. This text includes "Functions and Graphs in Applications" (Ch 0.6) which is fourteen pages of preparation for word problems. Authors of a book on
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s chose their exercises freely: :In order to enhance the attractiveness of this book as a
textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textbook ...
, we have included worked-out examples at appropriate points in the text and have included lists of exercises for Chapters 1 — 9. These exercises range from routine problems to alternative proofs of key theorems, but containing also material going beyond what is covered in the text. J. C. Maxwell explained how exercise facilitates access to the
language of mathematics The language of mathematics or mathematical language is an extension of the natural language (for example English) that is used in mathematics and in science for expressing results (scientific laws, theorems, proofs, logical deductions, etc) with ...
: :As mathematicians we perform certain mental operations on the symbols of number or quantity, and, proceeding step by step from more simple to more complex operations, we are enabled to express the same thing in many different forms. The equivalence of these different forms, though a necessary consequence of self-evident axioms, is not always, to our minds, self-evident; but the mathematician, who by long practice has acquired a familiarity with many of these forms, and has become expert in the processes which lead from one to another, can often transform a perplexing expression into another which explains its meaning in more intelligible language. The individual instructors at various colleges use exercises as part of their mathematics courses. Investigating
problem solving Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
in universities, Schoenfeld noted: :Upper division offerings for mathematics majors, where for the most part students worked on collections of problems that had been compiled by their individual instructors. In such courses emphasis was on learning by doing, without an attempt to teach specific heuristics: the students worked lots of problems because (according to the implicit instructional model behind such courses) that’s how one gets good at mathematics. Such exercise collections may be proprietary to the instructor and his institution. As an example of the value of exercise sets, consider the accomplishment of Toru Kumon and his Kumon method. In his program, a student does not proceed before mastery of each level of exercise. At the Russian School of Mathematics, students begin multi-step problems as early as the first grade, learning to build on previous results to progress towards the solution. In the 1960s, collections of mathematical exercises were translated from
Russian Russian(s) refers to anything related to Russia, including: *Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries * Rossiyane (), Russian language term for all citizens and p ...
and published by W. H. Freeman and Company: ''The USSR Olympiad Problem Book'' (1962), ''Problems in Higher Algebra'' (1965), and ''Problems in Differential Equations'' (1963).


History

In China, from ancient times counting rods were used to represent numbers, and arithmetic was accomplished with
rod calculus Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. ...
and later the
suanpan The suanpan (), also spelled suan pan or souanpan) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely ''Supplementary Notes on the Art of Figures'' written by Xu Yue. However, the exact design o ...
. The
Book on Numbers and Computation The ''Book on Numbers and Computation'' (), or the ''Writings on Reckoning'', is one of the earliest known Chinese mathematical treatises. It was written during the early Western Han dynasty, sometime between 202 BC and 186 BC.Liu et al. (2003), ...
and the
Nine Chapters on the Mathematical Art ''The Nine Chapters on the Mathematical Art'' () is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest su ...
include exercises that are exemplars of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
. In about 980
Al-Sijzi Abu Sa'id Ahmed ibn Mohammed ibn Abd al-Jalil al-Sijzi (c. 945 - c. 1020, also known as al-Sinjari and al-Sijazi; fa, ابوسعید سجزی; Al-Sijzi is short for " Al-Sijistani") was an Iranian Muslim astronomer, mathematician, and astrolo ...
wrote his ''Ways of Making Easy the Derivation of Geometrical Figures'', which was translated and published by
Jan Hogendijk Jan Pieter Hogendijk (born 21 July 1955) is a Dutch mathematician and historian of science. Since 2005, he is professor of history of mathematics at the University of Utrecht. Hogendijk became a member of the Royal Netherlands Academy of Arts a ...
in 1996. An Arabic language collection of exercises was given a Spanish translation as ''Compendio de Algebra de Abenbéder'' and reviewed in
Nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are p ...
. In Europe before 1900, the science of
graphical perspective Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
framed geometrical exercises. For example, in 1719 Brook Taylor wrote in ''New Principles of Linear Perspective'' :
he Reader He or HE may refer to: Language * He (pronoun), an English pronoun * He (kana), the romanization of the Japanese kana へ * He (letter), the fifth letter of many Semitic alphabets * He (Cyrillic), a letter of the Cyrillic script called ''He'' ...
will find much more pleasure in observing how extensive these Principles are, by applying them to particular Cases which he himself shall devise, while he exercises himself in this Art,... Taylor continued :...for the true and best way of learning any Art, is not to see a great many Examples done by another Person; but to possess ones self first of the Principles of it, and then to make them familiar, by exercising ones self in the Practice. The use of writing slates in schools provided an early format for exercises. Growth of exercise programs followed introduction of written examinations and study based on pen and paper.
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
described preparation for the
entrance examination An entrance examination is an examination that educational institutions conduct to select prospective students for admission. It may be held at any stage of education, from primary to tertiary, even though it is typically held at tertiary stage. ...
of
École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savo ...
as :...a course of "mathematiques especiales". This is an extraordinarily strong concentration of mathematical education – up to 16 hours a week – in which elementary analytic geometry and mechanics, and recently infinitesimal calculus also, are thoroughly studied and are made into a securely mastered tool by means of many exercises.
Sylvestre Lacroix Sylvestre can refer to: People Surname Given name Middle name * Carlos Sylvestre Begnis (1903–1980), Argentine medical doctor and politician * Philippe Sylvestre Dufour (1622–1687), French Protestant apothecary, banker, collector, ...
was a gifted teacher and expositor. His book on descriptive geometry uses sections labelled "Probleme" to exercise the reader’s understanding. In 1816 he wrote ''Essays on Teaching in General, and on Mathematics Teaching in Particular'' which emphasized the need to exercise and test: :The examiner, obliged, in the short-term, to multiply his questions enough to cover the subjects that he asks, to the greater part of the material taught, cannot be less thorough, since if, to abbreviate, he puts applications aside, he will not gain anything for the pupils’ faculties this way. Andrew Warwick has drawn attention to the historical question of exercises: :The inclusion of illustrative exercises and problems at the end of chapters in textbooks of mathematical physics is now so commonplace as to seem unexceptional, but it is important to appreciate that this pedagogical device is of relatively recent origin and was introduced in a specific historical context.Andrew Warwick (2003) ''Masters of Theory: Cambridge and the Rise of Mathematical Physics'',
University of Chicago Press The University of Chicago Press is the largest and one of the oldest university presses in the United States. It is operated by the University of Chicago and publishes a wide variety of academic titles, including ''The Chicago Manual of Style'', ...
In reporting
Mathematical tripos The Mathematical Tripos is the mathematics course that is taught in the Faculty of Mathematics at the University of Cambridge. It is the oldest Tripos examined at the University. Origin In its classical nineteenth-century form, the tripos was ...
examinations instituted at
Cambridge University , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...
, he notes :Such cumulative, competitive learning was also accomplished more effectively by private tutors using individual tuition, specially prepared manuscripts, and graded examples and problems, than it was by college lecturers teaching large classes at the pace of the mediocre. Explaining the relationship of examination and exercise, he writes :...by the 1830s it was the problems on examination papers, rather than exercises in textbooks, that defined the standard to which ambitious students aspired... ambridge studentsnot only expected to find their way through the merest sketch of an example, but were taught to regard such exercises as useful preparation for tackling difficult problems in examinations. Explaining how the reform took root, Warwick wrote: :It was widely believed in Cambridge that the best way of teaching mathematics, including the new analytical methods, was through practical examples and problems, and, by the mid-1830s, some of the first generation of young college fellows to have been taught higher analysis this way were beginning both to undertake their own research and to be appointed Tripos examiners. Warwick reports that in Germany,
Franz Ernst Neumann Franz Ernst Neumann (11 September 1798 – 23 May 1895) was a German mineralogist, physicist and mathematician. Biography Neumann was born in Joachimsthal, Margraviate of Brandenburg, near Berlin. In 1815 he interrupted his studies at Berlin to ...
about the same time "developed a common system of graded exercises that introduced student to a hierarchy of essential mathematical skills and techniques, and ...began to construct his own
problem set A problem set, sometimes shortened as pset, is a teaching tool used by many universities. Most courses in physics, math, engineering, chemistry, and computer science will give problem sets on a regular basis. They can also appear in other subje ...
s through which his students could learn their craft." In Russia,
Stephen Timoshenko Stepan Prokofyevich Timoshenko (russian: Степан Прокофьевич Тимошенко, p=sʲtʲɪˈpan prɐˈkofʲjɪvʲɪtɕ tʲɪmɐˈʂɛnkə; uk, Степан Прокопович Тимошенко, Stepan Prokopovych Tymoshenko; ...
reformed instruction around exercises. In 1913 he was teaching strength of materials at the
Petersburg State University of Means of Communication Emperor Alexander I St. Petersburg State Transport University (PGUPS) (russian: Петербургский государственный университет путей сообщения Императора Александра I, abbreviat ...
. As he wrote in 1968, : racticalexercises were not given at the Institute, and on examinations the students were asked only theoretical questions from the adopted textbook. I had to put an end to this kind of teaching as soon as possible. The students clearly understood the situation, realized the need for better assimilation of the subject, and did not object to the heavy increase in their work load. The main difficulty was with the teachers – or more precisely, with the examiners, who were accustomed to basing their exams on the book. Putting practical problems on the exams complicated their job. They were persons along in years...the only hope was to bring younger people into teaching.
Stephen Timoshenko Stepan Prokofyevich Timoshenko (russian: Степан Прокофьевич Тимошенко, p=sʲtʲɪˈpan prɐˈkofʲjɪvʲɪtɕ tʲɪmɐˈʂɛnkə; uk, Степан Прокопович Тимошенко, Stepan Prokopovych Tymoshenko; ...
(1968) ''As I Remember'', Robert Addis translator, pages 133,4, D. Van Nostrand Company


See also

*
Algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
* Worked-example effect


References


External links

*
Tatyana Afanasyeva Tatyana Alexeyevna Afanasyeva (russian: link=no, Татья́на Алексе́евна Афана́сьева) ( Kiev, 19 November 1876 – Leiden, 14 April 1964) (also known as Tatiana Ehrenfest-Afanaseva or spelled Afanassjewa) was a Russia ...
(1931
Exercises in Experimental Geometry
from
Pacific Institute for the Mathematical Sciences The Pacific Institute for the Mathematical Sciences (PIMS) is a mathematical institute created in 1996 by universities in Western Canada and the Northwestern United States to promote research and excellence in all areas of the mathematical science ...
. *
Vladimir Arnold Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–A ...
(2004
Exercises for students from age 5 to 15
a
IMAGINARY platform
*
James Alfred Ewing Sir James Alfred Ewing MInstitCE (27 March 1855 − 7 January 1935) was a Scottish physicist and engineer, best known for his work on the magnetic properties of metals and, in particular, for his discovery of, and coinage of the word, ''hys ...
(1911
Examples in Mathematics, Mechanics, Navigation and Nautical Astronomy, Heat and Steam, Electricity, for the use of Junior Officers Afloat
from
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
. * Jim Hefferon & others (2004) {{Mathematics education Mathematics education